Papers

Causal Dynamical Triangulations

The Universe from Scratch-Renate LollA fascinating and deep question about nature is what one would see if one could probe space and time at smaller and smaller distances. Already the 19th-century founders of modern geometry contemplated the possibility that a piece of empty space that looks completely smooth and structureless to the naked eye might have an intricate microstructure at a much smaller scale. Our vastly increased understanding of the physical world acquired during the 20th century has made this a certainty. The laws of quantum theory tell us that looking at spacetime at ever smaller scales requires ever larger energies, and, according to Einstein's theory of general relativity, this will alter spacetime itself: it will acquire structure in the form of "curvature". What we still lack is a definitive Theory of Quantum Gravity to give us a detailed and quantitative description of the highly curved and quantum-fluctuating geometry of spacetime at this so-called Planck scale. - This article outlines a particular approach to constructing such a theory, that of Causal Dynamical Triangulations, and its achievements so far in deriving from first principles why spacetime is what it is, from the tiniest realms of the quantum to the large-scale structure of the universe.

The Emergence of Spacetime, or, Quantum Gravity on Your Desktop-Renate LollIs there an approach to quantum gravity which is conceptually simple, relies on very
few fundamental physical principles and ingredients, emphasizes geometric (as opposed to
algebraic) properties, comes with a definite numerical approximation scheme, and produces
robust results, which go beyond showing mere internal consistency of the formalism? The
answer is a resounding yes: it is the attempt to construct a nonperturbative theory of
quantum gravity, valid on all scales, with the technique of so-called Causal Dynamical Triangulations.
Despite its conceptual simplicity, the results obtained up to now are far from
trivial. Most remarkable at this stage is perhaps the fully dynamical emergence of a classical
background (and solution to the Einstein equations) from a nonperturbative sum over geometries,
without putting in any preferred geometric background at the outset. In addition,
there is concrete evidence for the presence of a fractal spacetime foam on Planckian distance
scales. The availability of a computational framework provides built-in reality checks of the
approach, whose importance can hardly be overestimated.

Reconstructing the Universe-Jan AmbjørnWe provide detailed evidence for the claim that nonperturbative quantum gravity,
defined through state sums of causal triangulated geometries, possesses a
large-scale limit in which the dimension of spacetime is four and the dynamics of
the volume of the universe behaves semiclassically. This is a first step in reconstructing
the universe from a dynamical principle at the Planck scale, and at the
same time provides a nontrivial consistency check of the method of causal dynamical
triangulations. A closer look at the quantum geometry reveals a number
of highly nonclassical aspects, including a dynamical reduction of spacetime to
two dimensions on short scales and a fractal structure of slices of constant time.

Introductory Causal Dynamical Triangulation-Alex ForcierThis report aims to present the main ideas of Regge calculus necessary to understand the basic premise of CDT. Next, the main strategy of the CDT approach is introduced in general terms. The main focus of this report is the 2-D model of CDT. The section on the 2-D model closely follows a single paper. While the 4-D or even 3-D case will behave very differently from the 2-D model, 2-D CDT can be solved exactly, and as such offers a better introductory exposition of CDT's methods. Higher-dimensional CDT requires a lot of computer simulation, and lies outside the scope of this report.
All derivations carried out explicitly are the result of the author's independent work in attempting to find and prove how the results presented were obtained by CDT authors. Because these derivations were made explicit by the author, this paper can act as a guide to those who are new to CDT.

Dynamically Triangulating Lorentzian Quantum Gravity-Jan AmbjørnFruitful ideas on how to quantize gravity are few and far between. In this paper, we give a complete description of a recently introduced non-perturbative gravitational path integral whose continuum limit has already been investigated extensively in d less than 4, with promising results. It is based on a simplicial regularization of Lorentzian space-times and, most importantly, possesses a well-defined, non-perturbative Wick rotation. We present a detailed analysis of the geometric and mathematical properties of the discretized model in d=3,4. This includes a derivation of Lorentzian simplicial manifold constraints, the gravitational actions and their Wick rotation. We define a transfer matrix for the system and show that it leads to a well-defined self-adjoint Hamiltonian. In view of numerical simulations, we also suggest sets of Lorentzian Monte Carlo moves. We demonstrate that certain pathological phases found previously in Euclidean models of dynamical triangulations cannot be realized in the Lorentzian case.

Quantum gravity on a laptop: 1+1 Dimensional Causal Dynamical Triangulation simulaton-Norman S.IsraelThe quest for quantum gravity has been long and difficult. Causal Dynamical Triangulation is a new and straightforward approach to quantum gravity that recovers classical spacetime at large scales by enforcing causality at small scales. CDT combines quantum physics with general relativity in a Feynman sum-over-geometries and converts the sum into a discrete statistical physics problem. We solve this problem using a new Monte Carlo simulation to compute the spatial fluctuations of an empty universe with one space and one time dimensions. Our results compare favorably with theory and provide an accessible but detailed introduction to quantum gravity via a simulation that runs on a laptop computer.

Two Dimensional Causal Dynamical Triangulation-Norman S.IsraelIn this paper, a theory of quantum gravity called Causal Dynamical Triangulation (CDT) is
explored. The 1+1 dimensional universe is simulated in xCode. This paper explains CDT in
general and presents and explains the results of 1+1 dimensional simulations. The critical value of
the reduced cosmological constant was found to be 1 (it is taken to be dimensionless).

A discrete history of the Lorentzian path integral-Renate LollIn these lecture notes, I describe the motivation behind a recent formulation of
a non-perturbative gravitational path integral for Lorentzian (instead of the usual
Euclidean) space-times, and give a pedagogical introduction to its main features. At
the regularized, discrete level this approach solves the problems of (i) having a welldefined
Wick rotation, (ii) possessing a coordinate-invariant cutoff, and (iii) leading to
convergent sums over geometries. Although little is known as yet about the existence
and nature of an underlying continuum theory of quantum gravity in four dimensions,
there are already a number of beautiful results in
d = 2 and
d = 3 where continuum
limits have been found. They include an explicit example of the inequivalence of
the Euclidean and Lorentzian path integrals, a non-perturbative mechanism for the
cancellation of the conformal factor, and the discovery that causality can act as an
effective regulator of quantum geometry.

Locally Causal Dynamical Triangulations in Two Dimensions-Ben RuijlWe analyze the universal properties of a new two-dimensional quantum gravity model defined in terms of Locally Causal Dynamical Triangulations (LCDT). Measuring the Hausdorff and spectral dimensions of the dynamical geometrical ensemble, we find numerical evidence that the continuum limit of the model lies in a new universality class of two-dimensional quantum gravity theories, inequivalent to both Euclidean and Causal Dynamical Triangulations.

A Numerical Simulation in 1+1 Dimensions of Locally Causal Dynamical Triangulations-Ben RuijlCausal Dynamical Triangulation (CDT) has proven to be a viable candidate for
quantum gravity models. A natural generalization of CDT relaxes the foliation
constraint, allowing for more structures in the triangulation. The characteristics
of this Locally Causal Dynamical Triangulation (LCDT) model are currently
unknown, even in 1+1 dimensions. This thesis presents a detailed analysis of
LCDT in 1+1 dimensions and tries to find the characteristics of the theory using
Monte Carlo simulations. We present evidence that there are non-contractible
closed timelike loops in LCDT and we find that the spectral dimension is compatible
with CDT, but that Hausdorff dimension is not evidently so.

Fixing the Boundaries for Causal Dynamical Triangulations in 2+1 Dimensions-Jonah M. MillerIn this paper, we modify the discrete Regge action of Causal Dynamical Triangulations
(CDT) to account for fixed boundary conditions on the spacetime manifold and to include an
appropriate boundary term. We demonstrate that the resulting geometries agree with previous
results from CDT with periodic boundary conditions—and more importantly, reduce to classical
general relativity in the appropriate limit. We further propose a number of interesting
questions that fixed-boundaries CDT may be able to answer.

A first look at transition amplitudes in (2+1)-dimensional causal dynamical triangulations-Joshua H. Cooperman We study a lattice regularization of the gravitational path integral--causal dynamical triangulations--for (2+1)-dimensional Einstein gravity with positive cosmological constant in the presence of past and future spacelike boundaries of fixed intrinsic geometries. For spatial topology of a 2-sphere, we determine the form of the Einstein-Hilbert action supplemented by the Gibbons-Hawking-York boundary terms within the Regge calculus of causal triangulations. Employing this action we numerically simulate a variety of transition amplitudes from the past boundary to the future boundary. To the extent that we have so far investigated them, these transition amplitudes appear consistent with the gravitational effective action previously found to characterize the ground state of quantum spacetime geometry within the Euclidean de Sitter-like phase. Certain of these transition amplitudes convincingly demonstrate that the so-called stalks present in this phase are numerical artifacts of the lattice regularization, seemingly indicate that the quantization technique of causal dynamical triangulations differs in detail from that of the no-boundary proposal of Hartle and Hawking, and possibly represent the first numerical simulations of portions of temporally unbounded quantum spacetime geometry within the causal dynamical triangulations approach. We also uncover tantalizing evidence suggesting that Lorentzian not Euclidean de Sitter spacetime dominates the ground state on sufficiently large scales.

A second look at transition amplitudes in (2+1)-dimensional causal dynamical triangulations-Joshua H. CoopermanStudying transition amplitudes in (2+1)-dimensional causal dynamical triangulations, Cooperman and Miller discovered speculative evidence for Lorentzian quantum geometries emerging from its Euclidean path integral. On the basis of this evidence, Cooperman and Miller conjectured that Lorentzian de Sitter spacetime, not Euclidean de Sitter space, dominates the ground state of the quantum geometry of causal dynamical triangulations on large scales, a scenario akin to that of the Hartle-Hawking no-boundary proposal in which Lorentzian spacetimes dominate a Euclidean path integral. We argue against this conjecture: we propose a more straightforward explanation of their findings, and we proffer evidence for the Euclidean nature of these seemingly Lorentzian quantum geometries. This explanation reveals another manner in which the Euclidean path integral of causal dynamical triangulations behaves correctly in its semiclassical limit--the implementation and interaction of multiple constraints.

A Validation of Causal Dynamical Triangulations-Rajesh KommuThe Causal Dynamical Triangulation (CDT) approach to quantum gravity is a lattice approximation to the gravitational path integral. Developed by Ambjørn, Jurkiewicz and Loll, it has yielded some important results, notably the emergence of classical spacetime and short scale dimensional reduction. However, virtually all the results reported so far have been based on a single computer code. In this paper we present the first completely independent verification of the CDT algorithm, and report the successful reproduction of the emergence of classical spacetime and smooth reduction in the spectral dimension of the 2+1 and 3+1 dimensional spacetimes.

Dynamically Triangulating Lorentzian Quantum Gravity-Jan AmbjørnFruitful ideas on how to quantize gravity are few and far between. In this
paper, we give a complete description of a recently introduced non-perturbative
gravitational path integral whose continuum limit has already been investigated
extensively in d < 4, with promising results. It is based on a simplicial regularization
of Lorentzian space-times and, most importantly, possesses a well-defined,
non-perturbative Wick rotation. We present a detailed analysis of the geometric
and mathematical properties of the discretized model in
d = 3, 4. This includes a derivation of Lorentzian simplicial manifold constraints, the gravitational actions
and their Wick rotation. We define a transfer matrix for the system and show that
it leads to a well-defined self-adjoint Hamiltonian. In view of numerical simulations,
we also suggest sets of Lorentzian Monte Carlo moves. We demonstrate that certain
pathological phases found previously in Euclidean models of dynamical triangulations
cannot be realized in the Lorentzian case.

Causal Dynamical Triangulation of Quantum Gravity in Three Dimensions-J. Z. ZhangThe theory of causal dynamical triangulation in (2+1) dimensions
is studied through the help of Monte Carlo simulations. The basic ideas
behind this particular approach to quantum gravity are outlined in the
paper. The numerical setup and the results from the simulations are
presented and discussed. Based on the numerical results, the theory is
shown to possess a non-trivial classical limit. The effective dimension of
the emergent quantum universe is examined by means of measurements
of the fractal spectral dimension.

Making the case for causal dynamical triangulations-Joshua H. CoopermanThe aim of the causal dynamical triangulations approach is to define nonperturbatively a quantum theory of gravity as the continuum limit of a lattice-regularized model of dynamical geometry. My aim in this paper is to give a concise yet comprehensive, impartial yet personal presentation of the causal dynamical triangulations approach.

Spontaneous Dimensional Reduction in Short-Distance Quantum Gravity?-Steven CarlipSeveral lines of evidence suggest that quantum gravity at very short distances may behave effectively as a two-dimensional theory. I summarize these hints, and offer an additional argument based on the strong-coupling limit of the Wheeler-DeWitt equation. The resulting scenario suggests a novel approach to quantum gravity at the Planck scale.

Nonperturbative Quantum Gravity-Jan AmbjørnAsymptotic safety describes a scenario in which general relativity can be quantized as a conventional field theory, despite being nonrenormalizable when expanding it around a fixed background geometry. It is formulated in the framework of the Wilsonian renormalization group and relies crucially on the existence of an ultraviolet fixed point, for which evidence has been found using renormalization group equations in the continuum.
"Causal Dynamical Triangulations" (CDT) is a concrete research program to obtain a nonperturbative quantum field theory of gravity via a lattice regularization, and represented as a sum over spacetime histories. In the Wilsonian spirit one can use this formulation to try to locate fixed points of the lattice theory and thereby provide independent, nonperturbative evidence for the existence of a UV fixed point.
We describe the formalism of CDT, its phase diagram, possible fixed points and the "quantum geometries" which emerge in the different phases. We also argue that the formalism may be able to describe a more general class of Ho\v{r}ava-Lifshitz gravitational models.

CDT- an Entropic Theory of Quantum Gravity-Jan AmbjørnIn these lectures we describe how a theory of quantum gravity may be constructed in terms of a lattice formulation based on so-called causal dynamical triangulations (CDT). We discuss how the continuum limit can be obtained and how to define and measure diffeomorphism-invariant correlators. In four dimensions, which has our main interest, the lattice theory has an infrared limit which can be identified with de Sitter spacetime. We explain why this infrared property of the quantum spacetime is nontrivial and due to "entropic" effects encoded in the nonperturbative path integral measure. This makes the appearance of the de Sitter universe an example of true emergence of classicality from microscopic quantum laws. We also discuss nontrivial aspects of the UV behaviour, and show how to investigate quantum fluctuations around the emergent background geometry. Finally, we consider the connection to the asymptotic safety scenario, and derive from it a new, conjectured scaling relation in CDT quantum gravity.

De Sitter Universe from Causal Dynamical Triangulations without Preferred Foliation-Samo JordanWe present a detailed analysis of a recently introduced version of Causal Dynamical Triangulations (CDT) that does not rely on a distinguished time slicing. Focussing on the case of 2+1 spacetime dimensions, we analyze its geometric and causal properties, present details of the numerical set-up and explain how to extract "volume profiles". Extensive Monte Carlo measurements of the system show the emergence of a de Sitter universe on large scales from the underlying quantum ensemble, similar to what was observed previously in standard CDT quantum gravity. This provides evidence that the distinguished time slicing of the latter is not an essential part of its kinematical set-up.

Causal Dynamical Triangulations in Four Dimensions-Andrzej GörlichRecent results obtained within a non-perturbative approach to quantum gravity based on the method of four-dimensional Causal Dynamical Triangulations are described. The phase diagram of the model consists of three phases. In the physically most interesting phase, the time-translational symmetry is spontaneously broken. Calculations of expectation values required introducing procedures taking into account the inhomogeneity of configurations. It was shown that the dynamically emerged four-dimensional background geometry corresponds to a Euclidean de Sitter space and reveals no fractality at large distances. Measurements of the covariance matrix of scale factor fluctuations allowed to reconstruct the effective action, which remained in agreement with the discrete minisuperspace action. Values of the Hausdorff dimension and spectral dimension of three-dimensional spatial slices suggest their fractal nature, which was confirmed by a direct analysis of triangulation structure. The Monte Carlo algorithm used to obtain presented results is described.

Dimension and Dimensional Reduction in Quantum Gravity-Steven CarlipA number of very different approaches to quantum gravity contain a common thread, a hint that spacetime at very short distances becomes effectively two dimensional. I review this evidence, starting with a discussion of the physical meaning of "dimension" and concluding with some speculative ideas of what dimensional reduction might mean for physics.

Spontaneous Dimensional Reduction in Quantum Gravity-Steven CarlipHints from a number of different approaches to quantum gravity point to a phenomenon of "spontaneous dimensional reduction" to two spacetime dimensions near the Planck scale. I examine the physical meaning of the term "dimension" in this context, summarize the evidence for dimensional reduction, and discuss possible physical explanations.

The phase structure of Causal Dynamical Triangulations with toroidal spatial topology-Jan AmbjørnWe investigate the impact of topology on the phase structure of four-dimensional Causal Dynamical Triangulations (CDT). Using numerical Monte Carlo simulations we study CDT with toroidal spatial topology. We confirm existence of all four distinct phases of quantum geometry earlier observed in CDT with spherical spatial topology. We plot the toroidal CDT phase diagram and find that it looks very similar to the case of the spherical spatial topology.

Phenomenology of Causal Dynamical Triangulations-Jakub MielczarekThe four dimensional Causal Dynamical Triangulations (CDT) approach to quantum gravity is already more than ten years old theory with numerous unprecedented predictions such as non-trivial phase structure of gravitational field and dimensional running. Here, we discuss possible empirical consequences of CDT derived based on the two features of the approach mentioned above. A possibility of using both astrophysical and cosmological observations to test CDT is discussed. We show that scenarios which can be ruled out at the empirical level exist.

From Causal Dynamical Triangulations To Astronomical Observations-Jakub MielczarekThis letter discusses phenomenological aspects of dimensional reduction predicted by the Causal Dynamical Triangulations (CDT) approach to quantum gravity. The deformed form of the dispersion relation for the fields defined on the CDT space-time is reconstructed.

Spectral Dimension of the Universe-Jan AmbjørnWe measure the spectral dimension of universes emerging from nonperturbative quantum gravity, defined through state sums of causal triangulated geometries. While four-dimensional on large scales, the quantum universe appears two-dimensional at short distances. We conclude that quantum gravity may be "self-renormalizing" at the Planck scale, by virtue of a mechanism of dynamical dimensional reduction.

Renormalization Group Flow in CDT-Jan AmbjørnWe perform a first investigation of the coupling constant flow of the nonperturbative lattice model of four-dimensional quantum gravity given in terms of Causal Dynamical Triangulations (CDT). After explaining how standard concepts of lattice field theory can be adapted to the case of this background-independent theory, we define a notion of "lines of constant physics" in coupling constant space in terms of certain semiclassical properties of the dynamically generated quantum universe. Determining flow lines with the help of Monte Carlo simulations, we find that the second-order phase transition line present in this theory can be interpreted as a UV phase transition line if we allow for an anisotropic scaling of space and time.

Other Quantum Gravity approaches that I like!

Causal Sets Theory

Introduction to causal sets: an alternate view of spacetime structure-David D. ReidThis paper provides a thorough introduction to the causal set hypothesis aimed at students, and other interested persons, with some knowledge of general relativity and nonrelativistic quantum mechanics. I elucidate the arguments for why the causal set structure might be the appropriate structure for a theory of quantum gravity. The logical and formal development of a causal set theory as well as a few illuminating examples are also provided.

Causal sets and the deep structure of spacetime-Fay DowkerThe causal set approach to quantum gravity embodies the concepts of causality and discreteness. This article explores some foundational and conceptual issues within causal set theory.

Causal Sets: Discrete Gravity-Rafael D. SorkinThese are some notes in lieu of the lectures I was scheduled to give, but had to cancel at the
last moment. In some places, they are more complete, in others much less so, regrettably.
I hope they at least give a feel for the subject and convey some of the excitement felt at the
moment by those of us working on it.

Causal Sets Dynamics: Review & Outlook-Petros WalldenCausal sets is an approach to quantum gravity, where spacetime is replaced by a causal set. It is fundamentally discrete, and the causal relations between spacetime elements is the only structure that remains. A complete theory should have (i) kinematics (ii) dynamics and (iii) phenomenology. In this contribution we will explore the dynamical part of the theory, focusing on recent developments. We will analyse (a) classical dynamics of the causal set, (b) quantum dynamics of matter and fields on a classical causal set and finally (c) quantum dynamics of the causal set.

Loop Quantum Gravity

Loop Quantum Gravity-Carlo RovelliThe problem of describing the quantum behavior of gravity, and thus understanding quantum spacetime, is still open. Loop quantum gravity is a well-developed approach to this problem. It is a mathematically well-defined background-independent quantization of general relativity, with its conventional matter couplings. Today research in loop quantum gravity forms a vast area, ranging from mathematical foundations to physical applications. Among the most significant results obtained so far are: (i) The computation of the spectra of geometrical quantities such as area and volume, which yield tentative quantitative predictions for Planck-scale physics. (ii) A physical picture of the microstructure of quantum spacetime, characterized by Planck-scale discreteness. Discreteness emerges as a standard quantum effect from the discrete spectra, and provides a mathematical realization of Wheeler’s “spacetime foam” intuition. (iii) Control of spacetime singularities, such as those in the interior of black holes and the cosmological one. This, in particular, has opened up the possibility of a theoretical investigation into the very early universe and the spacetime regions beyond the Big Bang. (iv) A derivation of the Bekenstein-Hawking black-hole entropy. (v) Low-energy calculations, yielding n-point functions well defined in a background-independent context. The theory is at the roots of, or strictly related to, a number of formalisms that have been developed for describing background-independent quantum field theory, such as spin foams, group field theory, causal spin networks, and others. I give here a general overview of ideas, techniques, results and open problems of this candidate theory of quantum gravity, and a guide to the relevant literature.

Asymptotic Safety

Quantum Einstein Gravity-Martin ReuterWe give a pedagogical introduction to the basic ideas and concepts of the Asymptotic Safety program in Quantum Einstein Gravity. Using the continuum approach based upon the effective average action, we summarize the state of the art of the field with a particular focus on the evidence supporting the existence of the non-trivial renormalization group fixed point at the heart of the construction. As an application, the multifractal structure of the emerging space-times is discussed in detail. In particular, we compare the continuum prediction for their spectral dimension with Monte Carlo data from the Causal Dynamical Triangulation approach.

String Theory

String Theory and M-Theory-Katrin BeckerString theory is one of the most exciting and challenging areas of modern theoretical physics. This book guides the reader from the basics of string theory to recent developments. It introduces the basics of perturbative string theory, world-sheet supersymmetry, space-time supersymmetry, conformal field theory and the heterotic string, before describing modern developments, including D-branes, string dualities and M-theory. It then covers string geometry and flux compactifications, applications to cosmology and particle physics, black holes in string theory and M-theory, and the microscopic origin of black-hole entropy. It concludes with Matrix theory, the AdS/CFT duality and its generalizations. This book is ideal for graduate students and researchers in modern string theory, and will make an excellent textbook for a one-year course on string theory.

History papers of Quantum Gravity

Prima Facie Questions in Quantum Gravity-C. J. IshamThe long history of the study of quantum gravity has thrown up a complex web of ideas and approaches. The aim of this article is to unravel this web a little by analysing some of the {\em prima facie\/} questions that can be asked of almost any approach to quantum gravity and whose answers assist in classifying the different schemes. Particular emphasis is placed on (i) the role of background conceptual and technical structure; (ii) the role of spacetime diffeomorphisms; and (iii) the problem of time.

Structural Issues in Quantum Gravity-C. J. IshamA discursive, non-technical, analysis is made of some of the basic issues that arise in almost any approach to quantum gravity, and of how these issues stand in relation to recent developments in the field. Specific topics include the applicability of the conceptual and mathematical structures of both classical general relativity and standard quantum theory. This discussion is preceded by a short history of the last twenty-five years of research in quantum gravity, and concludes with speculations on what a future theory might look like.

An introduction to quantum gravity-Giampiero EspositoQuantum gravity was born as that branch of modern theoretical physics that tries to unify its guiding principles, i.e., quantum mechanics and general relativity. Nowadays it is providing new insight into the unification of all fundamental interactions, while giving rise to new developments in mathematics. The various competing theories, e.g. string theory and loop quantum gravity, have still to be checked against observations. We review the classical and quantum foundations necessary to study field-theory approaches to quantum gravity, the passage from old to new unification in quantum field theory, canonical quantum gravity, the use of functional integrals, the properties of gravitational instantons, the use of spectral zeta-functions in the quantum theory of the universe, Hawking radiation, some theoretical achievements and some key experimental issues.

Is Quantum Gravity Necessary?-Steven CarlipIn view of the enormous difficulties we seem to face in quantizing general relativity, we should perhaps consider the possibility that gravity is a fundamentally classical interaction. Theoretical arguments against such mixed classical-quantum models are strong, but not conclusive, and the question is ultimately one for experiment. I review some work in progress on the possibility of experimental tests, exploiting the nonlinearity of the classical-quantum coupling, that could help settle this question.

Quantum Gravity: General Introduction and Recent Developments-Claus KieferI briefly review the current status of quantum gravity. After giving some general motivations for the need of
such a theory, I discuss the main approaches in quantizing general relativity: Covariant approaches (perturbation
theory, effective theory, and path integrals) and canonical approaches (quantum geometrodynamics, loop quantum gravity). I then address quantum gravitational aspects of string theory. This is followed by a discussion of black holes and quantum cosmology. I end with some remarks on the observational status of quantum gravity

Cosmology

General relativity without coordinates-T. ReggeIn this paper we develop an approach to the theory of Riemannian manifolds which avoids the use of co-ordinates. Curved spaces are approximated by higher-dimensional analogs of polyhedra. Among the advantages of this procedure we may list the possibility of condensing into a simplified model the essential features of topologies like Wheeler’s wormhole and a deeper geometrical insight.

Renormalization and Effective Lagrangians-Joseph PolchinskiThere is a strong intuitive understanding of renormalization, due to Wilson, in terms of the scaling of effective lagrangians. We show that this can be made the basis for a proof of perturbative renormalization. We first study renormalizability in the language of renormalization group flows for a toy renormalization group equation. We then derive an exact renormalization group equation for a four-dimensional λ ø^(4) theory with a momentum cutoff. We organize the cutoff dependence of the effective lagrangian into relevant and irrelevant parts, and derive a linear equation for the irrelevant part. A lengthy but straightforward argument establishes that the piece identified as irrelevant actually is so in perturbation theory. This implies renormalizability. The method extends immediately to any system in which a momentum-space cutoff can be used, but the principle is more general and should apply for any physical cutoff. Neither Weinberg's theorem nor arguments based on the topology of graphs are needed.

Observation of Gravitational Waves from a Binary Black Hole Merger-B. P. Abbott et al.On September 14, 2015 at 09:50:45 UTC the two detectors of the Laser Interferometer Gravitational-Wave
Observatory simultaneously observed a transient gravitational wave signal. This is the first direct
detection of gravitational waves and the first observation of a binary black hole merger.

Feynman Quantization of General Relativity-Charles W. MisnerThis paper reports the beginnings of the quantum theory of general relativity based on the
Feynman integral or "sum over histories". We have not seen a way to use the Feynman integral to solve
immediately all the principal problems. We have to study the theory one piece at a time and to set each
fragment in place when we are able to understand it. In this sort of approach we need not follow any logical
order, but may study the easy parts first and hope to fill in the rest later. However, some over-all picture of
what the completed puzzle may look like is necessary in order to recognize the pieces.

Quantum Field Theory

Quantum Field Theory-Frank WilczekI discuss the general principles underlying quantum field theory, and attempt to identify its most profound consequences. The deepest of these consequences result from the infinite number of degrees of freedom invoked to implement locality. I mention a few of its most striking successes, both achieved and prospective. Possible limitations of quantum field theory are viewed in the light of its history.